A rational approximation of the arctangent function and a new approach in computing pi

TL;DR

This paper introduces a new rational approximation for the arctangent function, enabling a novel asymptotic formula for pi, based on series expansions related to the error function and Gaussian functions.

Contribution

It derives a new rational approximation of arctangent that generalizes to related functions, leading to an innovative asymptotic formula for pi.

Findings

Derived a rational approximation for arctangent

Developed a new asymptotic formula for pi

Generalized the approximation to related functions

Abstract

We have shown recently that integration of the error function ${\rm{erf}}\left( x \right)$ represented in form of a sum of the Gaussian functions provides an asymptotic expansion series for the constant pi. In this work we derive a rational approximation of the arctangent function $\arctan \left( x \right)$ that can be readily generalized it to its counterpart $ - {\rm{sgn}}\left( x \right)\pi /2 + \arctan \left( x \right)$, where ${\rm{sgn}}\left( x \right)$ is the signum function. The application of the expansion series for these two functions leads to a new asymptotic formula for $\pi$.